Find the Equation Given the Roots 3i and square root of 6

$3 i$ and $\sqrt{6}$

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

The root at $x = 3 i$ was found by solving for $x$ when $x - (3 i) = y$ and $y = 0$ .

The factor is $x - 3 i$

The root at $x = - 3 i$ was found by solving for $x$ when $x - (- 3 i) = y$ and $y = 0$ .

The factor is $x + 3 i$

The root at $x = \sqrt{6}$ was found by solving for $x$ when $x - (\sqrt{6}) = y$ and $y = 0$ .

The factor is $x - \sqrt{6}$

Combine all the factors into a single equation.

$y = (x - 3 i) (x + 3 i) (x - \sqrt{6})$

Multiply all the factors to simplify the equation $y = (x - 3 i) (x + 3 i) (x - \sqrt{6})$ .

Expand $(x - 3 i) (x + 3 i)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x + 3 i) - 3 i (x + 3 i)) (x - \sqrt{6})$

Apply the distributive property.

$y = (x \cdot x + x (3 i) - 3 i (x + 3 i)) (x - \sqrt{6})$

Apply the distributive property.

$y = (x \cdot x + x (3 i) - 3 i x - 3 i (3 i)) (x - \sqrt{6})$

Simplify terms.

Combine the opposite terms in $x \cdot x + x (3 i) - 3 i x - 3 i (3 i)$ .

Reorder the factors in the terms $x (3 i)$ and $- 3 i x$ .

$y = (x \cdot x + 3 i x - 3 i x - 3 i (3 i)) (x - \sqrt{6})$

Subtract $3 i x$ from $3 i x$ .

$y = (x \cdot x + 0 - 3 i (3 i)) (x - \sqrt{6})$

Add $x \cdot x$ and $0$ .

$y = (x \cdot x - 3 i (3 i)) (x - \sqrt{6})$

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} - 3 i (3 i)) (x - \sqrt{6})$

Multiply $- 3 i (3 i)$ .

Multiply $3$ by $- 3$ .

$y = (x^{2} - 9 i i) (x - \sqrt{6})$

Raise $i$ to the power of $1$ .

$y = (x^{2} - 9 (i i)) (x - \sqrt{6})$

Raise $i$ to the power of $1$ .

$y = (x^{2} - 9 (i i)) (x - \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = (x^{2} - 9 i^{1 + 1}) (x - \sqrt{6})$

Add $1$ and $1$ .

$y = (x^{2} - 9 i^{2}) (x - \sqrt{6})$

Rewrite $i^{2}$ as $- 1$ .

$y = (x^{2} - 9 \cdot - 1) (x - \sqrt{6})$

Multiply $- 9$ by $- 1$ .

$y = (x^{2} + 9) (x - \sqrt{6})$

Expand $(x^{2} + 9) (x - \sqrt{6})$ using the FOIL Method.

Apply the distributive property.

$y = x^{2} (x - \sqrt{6}) + 9 (x - \sqrt{6})$

Apply the distributive property.

$y = x^{2} x + x^{2} (- \sqrt{6}) + 9 (x - \sqrt{6})$

Apply the distributive property.

$y = x^{2} x + x^{2} (- \sqrt{6}) + 9 x + 9 (- \sqrt{6})$

Simplify terms.

Simplify each term.

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$y = x^{2} x + x^{2} (- \sqrt{6}) + 9 x + 9 (- \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{2 + 1} + x^{2} (- \sqrt{6}) + 9 x + 9 (- \sqrt{6})$

Add $2$ and $1$ .

$y = x^{3} + x^{2} (- \sqrt{6}) + 9 x + 9 (- \sqrt{6})$

Multiply $- 1$ by $9$ .

$y = x^{3} + x^{2} (- \sqrt{6}) + 9 x - 9 \sqrt{6}$

Reorder factors in $x^{3} + x^{2} (- \sqrt{6}) + 9 x - 9 \sqrt{6}$ .

$y = x^{3} - x^{2} \sqrt{6} + 9 x - 9 \sqrt{6}$

Find the Equation Given the Roots 3i and square root of 6

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